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A matter vertex with left and right oscillator levels $N,\tilde{N}$ and momentum $k$ has weights $N+{\alpha }^{\prime }{k}^2/4,\tilde{N}+{\alpha }^{\prime }{k}^2/4$. An integrated string vertex operator must have weights $(1,1)$, giving the displayed mass shell and level matching with $M^2=-k^2$. Physical states are further identified modulo null string states, equivalently through BRST cohomology; merely counting all weight-one primaries would include gauge redundancies.

In a two-dimensional conformal field theory, a quasi-primary operator of conformal weights $(h,\tilde{h})$ obeysfor global conformal, or Möbius, transformations. A primary operator obeys this law for every locally invertible holomorphic and antiholomorphic conformal coordinate change. Quasi-primary is therefore weaker: the stress tensor at nonzero central charge is quasi-primary, but its Schwarzian transformation term prevents it from being primary under a general map.

A primary state is a simultaneous eigenstate of $L_0,{\tilde{L}}_0$ with eigenvalues $h,\tilde{h}$, satisfyingA quasi-primary state need only satisfy the corresponding $n=1$ conditions. Negative modes of the Virasoro algebra generate descendants.

In radial quantization, a local operator at the origin prepares a state on a surrounding circle; logarithmic radius is Euclidean time. The state–operator correspondence is $|\mathcal{O}⟩=\mathcal{O}(0)|0⟩$. Acting with a Virasoro generator means inserting its stress-tensor contour around that operator. For a primary operator, the operator product expansion isMultiplication by $z^{n+1}$ and the contour residue give $L_0|\mathcal{O}⟩=h|\mathcal{O}⟩$, $L_{-1}|\mathcal{O}⟩=|\partial \mathcal{O}⟩$, and $L_n|\mathcal{O}⟩=0$ for $n>0$. Conversely the contour modes are the coefficients of the singular expansion; vanishing positive modes removes every pole of order three and higher. The remaining double and simple poles generate precisely the primary transformation law. The antiholomorphic argument is identical. This proves the state-operator correspondence for Virasoro primaries in both directions.

An integrated string vertex operator is integrated with $d^{2}z$. The measure transforms with weights $(-1,-1)$, so conformal invariance requires its matter operator to have weights $(1,1)$. In old covariant string quantization the same requirement is $L_{n>0}={\tilde{L}}_{n>0}=0$ and $L_0={\tilde{L}}_0=1$. The ghost-dressed unintegrated vertex $c\bar{c}V$ has total weights $(0,0)$. Physical states are identified modulo null string states, or more systematically by BRST cohomology; raw counting of all primary operators includes gauge redundancy.

For a free embedding boson the holomorphic stress tensor is $T=-:\partial X\cdot \partial X:/{\alpha }^{\prime }$. Contracting it with $:e^{ik\cdot X}:$ gives weight ${\alpha }^{\prime }{k}^2/4$; every holomorphic oscillator of level $r$ adds $r$ to that weight. ThusWith Lorentzian target signature, $M^2=-k^2$, so the closed-bosonic-string mass shell from primary weights isThe ground state is tachyonic, level one is massless, and higher levels are massive. The primary conditions also constrain the polarization tensors; the two zero-mode conditions fix the mass and level matching.

For the commuting beta-gamma system, write $r=z-w$. The displayed exchange leaves the arguments attached to their respective fields. Exchanging the arguments instead givesThis radial-ordering sign in commuting beta-gamma contractions must be kept when differentiating. Expanding the total derivative in the proposed stress tensor givesThe contraction $\gamma (z)\beta (w)\sim 1/r$ and its derivative $\partial \gamma (z)\beta (w)\sim -1/r^2$ giveSimilarly $\partial \beta (z)\gamma (w)\sim 1/r^2$ and $\beta (z)\gamma (w)\sim -1/r$ giveThere are no higher poles. Both fields are holomorphic and $\bar{T}=0$, so

To compute the central charge, put $A=:\partial \beta \gamma :$ and $B=:\beta \partial \gamma :$. Only double Wick contractions can give the fourth-order pole in $TT$. Their coefficients arewhere only the double-contraction terms are shown. For example, the mixed contraction uses $\partial \beta (z)\partial \gamma (w)\sim 2/r^3$ and $\gamma (z)\beta (w)\sim 1/r$. All Wick pairings have bosonic signs. Therefore the fourth-order coefficient isThe stress-tensor operator product expansion has fourth-order coefficient $c/2$, yieldingThis is the stress tensor of a commuting beta-gamma system calculation. Single contractions supply the usual $2T/r^2+\partial T/r$ terms, with the potential third-order terms canceling.

Use the free-boson operator product expansionWrite $E_p=:e^{ip\cdot X}:$ and $ϵ=z-w$. The needed Wick contractions are $\partial X^{\mu }(z)\partial X^{\nu }(w)\sim -{\alpha }^{\prime }{\eta }^{\mu \nu }/(2{ϵ}^2)$ and $\partial X^{\mu }(z)E_p(w)\sim -i{\alpha }^{\prime }{p}^{\mu }{E}_p(w)/(2ϵ)$. For $V_{\zeta }={\zeta }_{\mu }:\partial X^{\mu }{E}_p:$, the cross double contraction in the worldsheet stress tensor has a cubic pole: either of its two derivatives can contract with the derivative in $V_{\zeta }$, while the other contracts with $E_p$. Including the factor $-1/{\alpha }^{\prime }$ givesA primary operator has no cubic pole. Hence the first vertex is primary precisely whenThe antiholomorphic conformal weight is $\bar{h}={\alpha }^{\prime }{p}^2/4$. Primarity alone puts no mass-shell condition on $p$.

For $W_{\zeta }={\zeta }_{\mu \nu }:\partial X^{\mu }\bar{\partial }{X}^{\nu }{E}_p:$, the holomorphic cubic pole is $-i{\alpha }^{\prime }{p}^{\mu }{\zeta }_{\mu \nu }:\bar{\partial }{X}^{\nu }{E}_p:/(2{ϵ}^3)$. The antiholomorphic operator product expansion gives the analogous pole involving $p^{\nu }{\zeta }_{\mu \nu }$. Thus the transversality of derivative-exponential primary vertices requiresThere is no trace-free requirement for primarity. A physical integrated string vertex operator must additionally have conformal weights $(1,1)$; for this vertex that gives $p^2=0$.

Now keep the fermionic signs in the real chiral Majorana fermion calculation. Contracting the external $\psi (w)$ through $:\psi (z)\partial \psi (z):$ givesMultiplying by $-1/2$ and Taylor expanding $\psi (z)$ at $w$ producesThis proves that $\psi$ is a primary operator with weight $h_{\psi }=1/2$. To find the central charge, in $T(z)T(w)$ the two double-contraction pairings contribute, before the overall factor $1/4$,The single Wick contractions give $2T(w)/{ϵ}^2+\partial T(w)/ϵ$, soComparing the leading term with $c/(2{ϵ}^4)$ proves $c_{\psi }=1/2$, as in the free chiral Majorana fermion conformal field theory.

With four embedding bosons and the ordinary reparameterization bc ghost system, cancellation of the worldsheet Weyl anomaly in each chirality requiresThus there are 44 real left-moving and 44 real right-moving fermions, equivalently 44 two-dimensional nonchiral Majorana fermions. These are internal free fermions in a four-dimensional bosonic string. Adding internal matter to the bosonic string does not introduce local worldsheet supersymmetry or its superghosts.

For the requested polynomial string vertex operators, work in the unprojected Neveu–Schwarz sector. A full model also chooses spin structures and a consistent projection, which the central-charge condition alone does not specify. If the holomorphic and antiholomorphic oscillator contributions to the conformal weights are ${\ell }_L,{\ell }_R$, physicality requiresTherefore only equal levels at most one can contribute to the requested nonpositive bosonic string mass spectrum.

With zero fermion insertions, level zero gives the tachyon vertex operator $E_p$, a scalar with $M^2=-4/{\alpha }^{\prime }$. Level one gives the massless closed-string vertex operator ${\zeta }_{\mu \nu }:\partial X^{\mu }\bar{\partial }{X}^{\nu }{E}_p:$, with the transverse conditions above and the string-state gauge redundancy that removes longitudinal polarizations. The physical transverse polarization tensor decomposes into a symmetric trace-free graviton, an antisymmetric Kalb–Ramond field, and a scalar dilaton. In four spacetime dimensions these have respectively two, one, and one physical polarizations; the two-form can be dualized to a scalar.

With two fermion insertions there are two possibilities. One fermion in each chirality givesThese are $44^2=1936$ scalar tachyons before any projection. There is no spacetime derivative or spacetime polarization to impose a vector transversality condition.

Two different fermions in the same chirality form an antisymmetric fermion bilinear, a weight-one current $J^{ij}=:{\psi }^i{\psi }^j:$; equal-species undifferentiated bilinears vanish. Closed-string level matching then requires a weight-one bosonic derivative in the opposite chirality. The resulting string vertex operators areThey have $M^2=0$ and are gauge vectors, with $p\cdot A^{ij}=0$ and $A_{\mu }^{ij}\sim A_{\mu }^{ij}+p_{\mu }{\lambda }^{ij}$, and likewise for $\tilde{A}$. Each chirality supplies $\left(\genfrac{}{}{0ex}{}{44}{2}\right)=946$ gauge vectors. The fermion bilinears generate the two $\mathfrak{so}(44)$ current algebras associated with the Special orthogonal Lie algebra, giving the unprojected gauge group $SO(44{)}_L\times SO(44{)}_R$. The tempting vertex $:{\psi }^i{\psi }^j{E}_p:$ alone fails closed-string level matching because its weights are $(1,0)$ before the common momentum contribution. Additional derivatives raise the levels above the nonpositive-mass range; a product of a left and a right bilinear has four fermion insertions and is outside this requested sector. This completes the zero- and two-fermion vertices of an internally fermionized bosonic string; a specified global projection can remove some of the listed states.

Choose a Euclidean string worldsheet and the usual normalization of the Polyakov action. For a closed oriented worldsheet $\Sigma$, the string nonlinear sigma model isHere ${\epsilon }^{12}=1$ is the alternating density, not the tensor with components $1/\sqrt{g}$. The imaginary coefficient in the Kalb–Ramond field term is the Euclidean continuation; in Lorentzian signature this term is real. $G$ is the spacetime metric and $\Phi$ the dilaton.

All three terms have worldsheet diffeomorphism invariance, and are covariant under spacetime coordinate transformations when $X$, $G$, $B$ and $\Phi$ are transformed together. Under a Weyl transformation $g_{ab}\mapsto e^{2\omega }{g}_{ab}$, $\sqrt{g}{g}^{ab}$ is invariant in two dimensions, so the metric coupling has classical Weyl invariance. The antisymmetric coupling is independent of the worldsheet metric, so it too has classical Weyl invariance.

The curvature coupling requires care. An infinitesimal Weyl transformation gives ${\delta }_{\omega }(\sqrt{g}{R}^{(2)})=-2\sqrt{g}{\Delta }_g\omega$. Integration by parts on the closed worldsheet therefore gives the Weyl variation of a nonconstant dilaton coupling:A constant dilaton coupling is topological and Weyl invariant; a general varying dilaton coupling is not separately classically Weyl invariant. Its classical variation participates in canceling the quantum worldsheet Weyl anomaly, producing the improved background equations below.

To relate these background couplings to the flat-space bosonic string mass spectrum, expand $G_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }(X)$ and express $h$ and $B$ as superpositions of plane waves. The perturbation of the action is a sum of integrated string vertex operators of the formThis is the vertex of ${\alpha }_{-1}^{\mu }{\tilde{\alpha }}_{-1}^{\nu }|p⟩$ in the massless first closed-string level. Its conformal weights are $(1+{\alpha }^{\prime }{p}^2/4,1+{\alpha }^{\prime }{p}^2/4)$, so marginality gives $p^2=0$. The primary operator conditions give transversality, and the string-state gauge redundancy removes longitudinal components. The symmetric trace-free polarization tensor is the graviton and perturbs $G$; its antisymmetric part is the Kalb–Ramond field and perturbs $B$. The scalar physical polarization is the dilaton. At the level of background fields its vertex involves the metric trace and the curvature improvement, so an arbitrary off-shell metric trace should not simply be identified with the physical dilaton without this distinction.

The spacetime gauge invariance of the antisymmetric coupling isIndeed its action is $S_B=i(2\pi {\alpha }^{\prime }{)}^{-1}{\int }_{\Sigma }{X}^{\ast }B$. Its variation isfor a closed worldsheet, by Stokes theorem. In components this is the total derivative ${\partial }_a({\epsilon }^{ab}{\Lambda }_{\nu }(X){\partial }_b{X}^{\nu })$: the term containing ${\partial }_a{\partial }_b{X}^{\nu }$ vanishes by antisymmetry. Thus only the Kalb-Ramond field strength $H$ can enter the local spacetime equations. With boundaries, a compensating boundary gauge-field transformation is needed.

Separate the dilaton into a constant ${\Phi }_0$ and a varying part. The Gauss-Bonnet theorem givesfor a connected closed orientable genus-$g$ worldsheet. Its Euclidean path-integral weight containsConsequently the sum over worldsheet topologies is the string genus expansion ${\sum }_{g\ge 0}{g}_s^{2g-2}{\mathcal{A}}_g$, with each coefficient integrating over that genus's worldsheet moduli. Each extra handle costs $g_s^2$, which is the loop-counting factor of string perturbation theory. With the usual normalization of $n$ external closed-string vertices, the connected amplitude carries $g_s^{2g-2+n}$. This dilaton Euler-characteristic weighting is distinct from the expansion in ${\alpha }^{\prime }$ of a fixed-worldsheet string nonlinear sigma model.

Finally quantize that two-dimensional theory. The background fields are its couplings. The background field expansion of a string sigma model, $X=x+\xi$ in target-space normal coordinates, contains curvature vertices quadratic in the fluctuation $\xi$. Contracting their two fluctuation indices produces the Ricci tensor multiplying $\partial x^{\mu }\bar{\partial }{x}^{\nu }$. Logarithmic short-distance contractions renormalize $G$, $B$, and $\Phi$, giving functional sigma-model beta functions. Gauge independence requires vanishing of the worldsheet Weyl anomaly, including the classical improvement from the varying dilaton, not merely an arbitrary choice of renormalization scale.

For clarity, in conventional leading-order normalization the improved coefficients areHere $H^2=H_{\mu \nu \rho }{H}^{\mu \nu \rho }$. The constant term in the dilaton coefficient is the matter-plus-ghost central charge deficit. At $D=26$, setting these coefficients to zero gives the leading spacetime field equationsFor the last equation, take the trace of the first, $R-H^2/4+2{\nabla }^2\Phi =0$, and combine it with ${\overline{\beta }}^{\Phi }=0$; this verifies the coefficients and signs. These are equivalently the equations from the leading string-frame massless effective actionHigher worldsheet orders give higher-derivative ${\alpha }^{\prime }$ corrections, whereas additional handles give string-loop corrections controlled by $g_s$. Thus quantum consistency of a two-dimensional string nonlinear sigma model determines the dynamics of its spacetime background fields. These leading coefficient conventions agree with the primary string-theory lectures' background-field calculation.

Use a mostly-plus target Minkowski spacetime metric and restore the string tension by $T=(2\pi {\alpha }^{\prime }{)}^{-1}$. After Wick rotation, the Polyakov action on a Riemann surface isThe Polyakov path integral integrates over embeddings and worldsheet metrics, dividing by worldsheet diffeomorphisms and Weyl transformations. For a consistent flat bosonic string theory, take the critical dimension of the bosonic string $d=26$. Conformal gauge turns the embedding fields into free scalars; its Faddeev-Popov determinant is represented by the bc system. The remaining integrations are over worldsheet moduli and external insertion points.

A tachyon vertex operator isThe worldsheet Green function is $⟨X^a(z)X^b(w)⟩=-({\alpha }^{\prime }/2){\eta }^{ab}\log |z-w{|}^2$. The operator product expansion with the holomorphic stress-energy tensor gives conformal weights $({\alpha }^{\prime }{k}^2/4,{\alpha }^{\prime }{k}^2/4)$. An integrated string vertex operator must have weights $(1,1)$, so the mass-shell condition is $k^2=4/{\alpha }^{\prime }$, or tachyonic mass parameter $m^2=-4/{\alpha }^{\prime }$.

At lowest order in the string coupling, the worldsheet is a Riemann sphere. For four external states, three insertion points can be fixed by a Möbius transformation. In the worldsheet ghost field description these three vertices carry $c\bar{c}$, while the fourth is integrated. The three-point worldsheet ghost correlator gives the squared product of the three fixed-point separations, precisely the sphere gauge fixing for four string vertices factor. Thus fixing three points is a gauge choice, rather than deleting three integrations without their Jacobian.

The integral over the constant embedding worldsheet zero mode gives $(2\pi {)}^d{\delta }^{(d)}({\sum }_i{k}_i)$. The remaining Gaussian integral, using normal ordering to remove self-contractions, gives the Koba-Nielsen factorSet $(z_1,z_2,z_3,z_4)=(0,z,1,\infty )$; the vertex at infinity is defined with its conformal-weight factor. Let all momenta be incoming and defineThe mass-shell condition and momentum conservation imply $s+t+u=-16/{\alpha }^{\prime }$. Since ${\alpha }^{\prime }{k}_1\cdot k_2=-4-{\alpha }^{\prime }s/2$ and similarly for the pair $(2,3)$, the sphere tachyon position integral becomesHere $\mathcal{N}$ depends on the normalization of external string vertex operators; it does not affect the kinematic dependence.

To evaluate the remaining worldsheet modulus, put $a=-1-{\alpha }^{\prime }s/4$ and $b=-1-{\alpha }^{\prime }t/4$. The complex beta integral isFor example, this identity follows by writing the two powers as Schwinger-parameter Gamma integrals, doing the two-dimensional Gaussian integral, and changing the two positive parameters to their sum and ratio. The complex beta integral first holds for $\mathrm{Re}a,\mathrm{Re}b>0$ and $\mathrm{Re}(a+b)<1$; the scattering result is its analytic continuation. The third gamma-function argument is $1-a-b=-1-{\alpha }^{\prime }u/4$. Absorbing $\pi$ into $\mathcal{N}$, the result is the tree-level four-tachyon amplitude, the Virasoro–Shapiro amplitude:It is symmetric in the three Mandelstam variables and has generic poles at $q=4(N-1)/{\alpha }^{\prime }$, matching the bosonic string mass spectrum. The formula is the leading term of string perturbation theory. Higher orders are constructed by the same Polyakov path integral prescription on higher-genus worldsheets, with four marked points and the corresponding ghost and worldsheet moduli measure; genus $g$ has coupling power $g_s^{2g+2}$.

The matter part of the massless closed-string vertex operator isIts symmetric trace-free polarization is the graviton, while its antisymmetric polarization is the Kalb–Ramond field, or B-field. The mass-shell condition $k^2=0$ and transversality make $V_{ϵ}$ a primary operator of conformal weights $(1,1)$, as required for an integrated string vertex operator.

The graviton polarization has the linearized gauge redundancywhile the antisymmetric polarization obeysIn either case the change in the integrated vertex is a worldsheet total derivative, hence vanishes on a closed worldsheet; in covariant language it is BRST-exact. This string-state gauge redundancy is the vertex-operator form of linearized target-space diffeomorphism or two-form gauge invariance.